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# Class Note for CSCI 124 with Professor Vora at GW (10)

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This 2 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/07/15

CSCI 124224 Discrete Structures 11 Existence of Units in Zm Poorvi L Vora In this module we show that an element in a nite ring has a multiplicative inverse if and only if it is not a zero divisor that an element 1 E Zm is a zero divisor if and only if gcdzTn y l and hence that an element 1 E Zm is a unit if and only if gcdzTn 1 Recall that A gtB gtB gtA gtA gtB gtB gtA where X denotes the negation of X Hence to show that an element in a nite ring has a multiplicative inverse if and only if it is not a zero divisor we show the converse which is equivalent Theorem An element 1 E R where R is a nite ring is a zero divisor if and only if it is not a unit Proof gt Suppose I is a zero divisor 3y 6 R y y 2 where 2 is the additive identity in R such that my 2 If I is a unit 3i 6 R such that ix u where u E R is the multiplicative identity Hence zy2 izyi2 y2 as T2 2 VT 6 R this has been done in class previously and is shown in the text This is a contradiction hence I is not a zero divisor 5 Suppose I is not a unit This implies that there is no y E R such that my u Now consider the value of IT for all values of T E R such that T y 2 Let the size of the nite ring R be TL Then there are TL 7 1 values of IT If these values are all distinct then one of these values has to be u because there are exactly TL 7 1 possibilities in R if 2 is not included and one of these is u If zT u then T is the multiplicative inverse of I But I is not a unit hence we cannot assume that all values of 17 are distinct Suppose 1 7 1 1 7 2 for 7 1 y 7 2 Then 1 7 1 7 1 7 2 2 and 17 1 7 7 2 2 as 7 1 y 7 2 1 is a zero divisor Theorem An element 1 E Zm is a zero divisor if and only if gcdz Tn y 1 Proof gt Suppose I is a zero divisor 3y 6 Zm such that my 0 Tnod Tn and y y 0 Tnod Tn That is and Tn does not divide y Hence some factor of Tn divides z and another divides y That is gcdz Tn y l CSCI 124 and CSCI 224VoIaGWU 2 5 Suppose gcdzm g y 1 Then I mg andm n29 Wherel lt nhng lt m and mm ngnlg ngg 0 mod m Hence 3y n2 y 0 mod m such that my 0 mod m and z is a zero divisor And the nal result simply puts the two results above together to get Theorem An element 1 E Zm is a unit if and only if gcdzm 1 Proof By the rst theorem 1 E Zm is a unit if and only if it is not a zero divisor By the second theorem it is not a zero divisor if and only if gcdz7 m 1 Hence the result

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